Mathematics for the interested outsider

Inducing the Trivial Representation

We really should see an example of inducing a representation. One example we’ll find extremely useful is when we start with the trivial representation.

So, let be a group and be a subgroup. Since this will be coming up a bunch, let’s just start writing for the trivial representation that sends each element of to the matrix . We want to consider the induced representation .

Well, we have a matrix representation, so we look at the induced matrix representation. We have to pick a transversal for the subgroup in . Then we have the induced matrix in block form:

In this case, each “block” is just a number, and it’s either or , depending on whether is in or not. But if , then latex g(t_jH)=(t_iH)$. That is, this is exactly the coset representation of corresponding to . And so all of these coset representations arise as induced representations.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.